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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series
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by Walter Gautschi and Gradimir V. Milovanović PDF
Math. Comp. 44 (1985), 177-190 Request permission


Polynomials ${\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda )$, $k = 0,1,2, \ldots$, are constructed which are orthogonal with respect to the weight distributions $d\lambda (t) = {(t/({e^t} - 1))^r}\;dt$ and $d\lambda (t) = {(1/({e^t} + 1))^r}\;dt$, $r = 1,2$, on $(0,\infty )$. Moment-related methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients ${\alpha _k},{\beta _k}$ in the recursion formula ${\pi _{k + 1}}(t) = (t - {\alpha _k}){\pi _k}(t) - {\beta _k}{\pi _{k - 1}}(t)$, $k = 0,1,2, \ldots$, ${\pi _0}(t) = 1$, ${\pi _{ - 1}}(t) = 0$. The discretization is effected by the Gauss-Laguerre and a composite Fejér quadrature rule, respectively. Numerical values of ${\alpha _k},{\beta _k}$, as well as associated error constants, are provided for $0 \leqslant k \leqslant 39$. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of n-point formulae, $n = 5(5)40$, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series.
    J. S. Blakemore, Solid State Physics, 2nd ed., Saunders, Philadelphia, Pa., 1974. W. Gautschi, "Algorithm 542—Incomplete gamma functions," ACM Trans. Math. Software, v. 5, 1979, pp. 482-489.
  • Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
  • Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, DOI 10.1137/0903018
  • J. F. Hart et al., Computer Approximations, Wiley, New York-London-Sydney, 1968. A. McLellan IV, "Tables of the Riemann zeta function and related functions," Math. Comp., v. 22, 1968, Review 69, pp. 687-688. S. S. Mitra & N. E. Massa, "Lattice vibrations in semiconductors," Chapter 3 in: Band Theory and Transport Properties (W. Paul, ed.), pp. 81-192. Handbook on Semiconductors (T. S. Moss, ed.), Vol. 1. North-Holland, Amsterdam, 1982. F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965. R. A. Smith, Semiconductors, 2nd ed., Cambridge Univ. Press, Cambridge, 1978. R. J. Van Overstraeten, H. J. DeMan & R. P. Mertens, "Transport equations in heavy doped silicon," IEEE Trans. Electron Dev., v. ED-20, 1973, pp. 290-298.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Math. Comp. 44 (1985), 177-190
  • MSC: Primary 65D32; Secondary 33A65, 65A05, 65B10, 81-08, 82-08
  • DOI:
  • MathSciNet review: 771039